Tauberian theorem for m‐spherical transforms on the Heisenberg group

In this paper we prove a Tauberian type theorem for the space L $ ^1 _{\bf m} $ ( H n ). This theorem gives sufficient conditions for a L $ ^1 _{\bf 0} $ ( H n ) submodule J ⊂ L $ ^1 _{\bf m} $ ( H n ) to make up all of L $ ^1 _{\bf m} $ ( H n ). As a consequence of this theorem, we are able to improve previous results on the Pompeiu problem with moments on the Heisenberg group for the space L ∞ ( H n ). In connection with the Pompeiu problem, given the vanishing of integrals ∫   | z |= r   iz m L g f ( z , 0) dσ ( z ) = 0 for all g ∈ H n and i = 1, 2 for appropriate radii r 1 and r 2 , we now have the (improved) conclusion $ {\bar {\bf Z}}^{\bf m} $ f ≡ 0, where $ {\bar {\bf Z}}^{\bf m} $ = $ \bar Z^{m_1}_1 $ · · · $ \bar Z^{m_n}_n $ and $ \bar Z_j $ form the standard basis for T (0,1) ( H n ). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)